Li's Family Has A Coupon For $\$49$ Off A Stay At Any Hotel. They Do Not Want To Spend More Than $\$150$ In All. An Inequality Representing This Situation Is $x - 49 \leq 150$.Explain How You Would Graph This Inequality. What

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Introduction

Linear inequalities are mathematical expressions that contain a variable and a constant, connected by a mathematical operation such as greater than, less than, greater than or equal to, or less than or equal to. In this article, we will explore how to graph a linear inequality, using the example of Li's family coupon for a hotel stay.

The Inequality

Li's family has a coupon for $49\$49 off a stay at any hotel. They do not want to spend more than $150\$150 in all. An inequality representing this situation is x−49≤150x - 49 \leq 150. This inequality can be rewritten as x≤199x \leq 199, where xx represents the total cost of the hotel stay.

Graphing the Inequality

To graph the inequality x≤199x \leq 199, we need to understand the concept of a number line. A number line is a line that represents all the real numbers, with positive numbers to the right of zero and negative numbers to the left of zero.

Step 1: Draw the Number Line

Draw a number line with a scale that includes all the numbers from −100-100 to 200200. This will give us a clear representation of the inequality.

Step 2: Identify the Boundary

The boundary of the inequality is the value of xx that makes the inequality true. In this case, the boundary is x=199x = 199. To identify the boundary, we need to find the point on the number line that corresponds to x=199x = 199.

Step 3: Shade the Region

Once we have identified the boundary, we need to shade the region that satisfies the inequality. Since the inequality is x≤199x \leq 199, we need to shade all the points to the left of x=199x = 199.

Step 4: Add an Arrow

To indicate that the inequality is less than or equal to, we need to add an arrow to the left of the boundary. This arrow indicates that all the points to the left of x=199x = 199 satisfy the inequality.

The Graph

The graph of the inequality x≤199x \leq 199 is a closed circle at x=199x = 199, with an arrow pointing to the left. This indicates that all the points to the left of x=199x = 199 satisfy the inequality.

Interpretation

The graph of the inequality x≤199x \leq 199 represents the situation where Li's family does not want to spend more than $150\$150 in all. The closed circle at x=199x = 199 indicates that the maximum cost is $199\$199, and the arrow pointing to the left indicates that all the points to the left of x=199x = 199 satisfy the inequality.

Conclusion

In this article, we have explored how to graph a linear inequality using the example of Li's family coupon for a hotel stay. We have learned how to draw a number line, identify the boundary, shade the region, and add an arrow to indicate the direction of the inequality. The graph of the inequality x≤199x \leq 199 represents the situation where Li's family does not want to spend more than $150\$150 in all.

Real-World Applications

Linear inequalities have many real-world applications, including finance, economics, and engineering. For example, a company may have a budget constraint that requires them to spend no more than a certain amount on a project. A linear inequality can be used to represent this constraint and determine the maximum amount that can be spent.

Tips and Tricks

When graphing a linear inequality, it is essential to remember the following tips and tricks:

  • Draw a number line with a scale that includes all the numbers from −100-100 to 200200.
  • Identify the boundary by finding the point on the number line that corresponds to the value of xx.
  • Shade the region that satisfies the inequality.
  • Add an arrow to indicate the direction of the inequality.

By following these tips and tricks, you can graph linear inequalities with ease and apply them to real-world problems.

Common Mistakes

When graphing linear inequalities, it is essential to avoid the following common mistakes:

  • Drawing the number line with an incorrect scale.
  • Identifying the boundary incorrectly.
  • Shading the region incorrectly.
  • Adding an arrow in the wrong direction.

By avoiding these common mistakes, you can ensure that your graph accurately represents the linear inequality.

Conclusion

Q: What is a linear inequality?

A: A linear inequality is a mathematical expression that contains a variable and a constant, connected by a mathematical operation such as greater than, less than, greater than or equal to, or less than or equal to.

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you need to follow these steps:

  1. Draw a number line with a scale that includes all the numbers from −100-100 to 200200.
  2. Identify the boundary by finding the point on the number line that corresponds to the value of xx.
  3. Shade the region that satisfies the inequality.
  4. Add an arrow to indicate the direction of the inequality.

Q: What is the boundary of a linear inequality?

A: The boundary of a linear inequality is the value of xx that makes the inequality true. It is the point on the number line that corresponds to the value of xx.

Q: How do I identify the boundary of a linear inequality?

A: To identify the boundary of a linear inequality, you need to find the point on the number line that corresponds to the value of xx. This can be done by solving the equation that is formed by setting the inequality to zero.

Q: What is the significance of the arrow in a linear inequality graph?

A: The arrow in a linear inequality graph indicates the direction of the inequality. If the inequality is greater than or equal to, the arrow points to the right. If the inequality is less than or equal to, the arrow points to the left.

Q: Can I graph a linear inequality with a negative coefficient?

A: Yes, you can graph a linear inequality with a negative coefficient. The graph will be a closed circle at the boundary, with an arrow pointing in the opposite direction of the inequality.

Q: How do I graph a linear inequality with a fraction?

A: To graph a linear inequality with a fraction, you need to follow these steps:

  1. Multiply both sides of the inequality by the denominator to eliminate the fraction.
  2. Graph the resulting linear inequality.
  3. Add an arrow to indicate the direction of the inequality.

Q: Can I graph a linear inequality with a decimal?

A: Yes, you can graph a linear inequality with a decimal. The graph will be a closed circle at the boundary, with an arrow pointing in the opposite direction of the inequality.

Q: How do I graph a system of linear inequalities?

A: To graph a system of linear inequalities, you need to follow these steps:

  1. Graph each inequality separately.
  2. Find the intersection of the two graphs.
  3. Shade the region that satisfies both inequalities.

Q: What is the significance of graphing a system of linear inequalities?

A: Graphing a system of linear inequalities is useful in solving real-world problems that involve multiple constraints. It helps to visualize the solution set and make informed decisions.

Conclusion

In conclusion, graphing linear inequalities is a crucial skill in mathematics and has many real-world applications. By following the steps outlined in this article, you can graph linear inequalities with ease and apply them to real-world problems. Remember to draw a number line with a scale that includes all the numbers from −100-100 to 200200, identify the boundary by finding the point on the number line that corresponds to the value of xx, shade the region that satisfies the inequality, and add an arrow to indicate the direction of the inequality. By following these tips and tricks, you can become proficient in graphing linear inequalities and apply them to real-world problems.